(This question has originally been posted on reddit, but I thought, that the question raised in the post above, might fit as well here on MO.)

I wanted to share with you something I stumbled upon about Hilbert spaces and Newton's formula for gravitation. Newton's formula for gravitation is represented by $$F(a,b) = G * m_a * m_b / | x_a - x_b|^2.$$

However, this can be equivalently expressed as

$$k(a,b) := \exp( -|x_a-x_b|^2/G ) = \exp( - m_a * m_b / F(a,b) )$$

which is known as the radial basis function kernel (RBF kernel) and has applications in machine learning and statistics. Interestingly, $k(a,b)$ can be shown to be the inner product $<a,b>$ to an infinite dimensional Hilbert space, specifically the Hilbert Sphere: $S_infty = \{ x \in H | k(x,x) = <x,x> = 1 \}$. By solving the last equality for $F(a,b)$, we can define the force in any Hilbert sphere with inner product $k(a,b)$ for $a \neq b$ as:

$$F(a,b) := - mu(a) * mu(b) / \log( k(a,b) )$$.

Now, if we consider only kernels that satisfy the axioms of force that implement Mach's principle (A. Assis' axioms), we could create something I would like to name Weber-Assis-Mach-space (WAM-space for short) that adds more dynamics to the Hilbert sphere. The axioms from A.Assis are as follows:

(A) Force is a vectorial quantity.

(B) The force that a material body A exerts on a material body B is equal and opposite to the force that B exerts on A.

(C) The sum of all forces on any material body is zero.

However, I am unsure if the Weber law for gravity, as described by Assis, allows one to define such a kernel. Additionally, from a physics standpoint, the geodesic distance in Hilbert sphere is given by $arccos(k(a,b))$, which results in

$$d(a,b) = arccos( \exp( - |x_a-x_b|^2/ G ) ) = arccos( \exp( - m_a * m_b / F(a,b) ) )$$.

I am imagening a so called WAM-space, where you can define objects having $a,b,c,d... $living in a Hilbert sphere with inner product k(a,b) having a certain 'mass' $mu(a), mu(b),...$ which are non-zero real numbers, and between two of these objects, there is a force:

$$F(a,b) := - mu(a) * mu(b) / log( k(a,b) )$$.

subject to the axioms of A.Assis, which can be found here in his book:

https://www.ifi.unicamp.br/~assis/Relational-Mechanics-Mach-Weber.pdf

I'm not entirely sure how this Hilbert space hypothesis could be tested experimentally, or if it's just a crazy idea, but I wanted to share it with the community and see where it leads. Let me know what you think!

This approach described above, might maybe be used to find a Hilbert sphere, where the forces of Newtons gravity and Coulombs electricity are combined to one force:

Suppose on the material body a are two forces of different quality such as the gravity force of Newton and the electric force of Coulomb. In physics, if they act upon the same body, then the sum of the forces does again by Newtons law, corresponds to a new force, hence:

$$F(a,b) = F_N(a,b) + F_C(a,b)$$ (*)

where $F_N$ is the Newton gravity force, and $F_C$ is the electric Coulomb force. The "masses" or let us better say, "quantities" are given by:

$$m(a) = (m_N(a),m_C(a)), m(b) = (m_N(b),m_C(b))$$

where $m_N(x)$ denotes the mass of $x$, $m_C(x)$ denotes the electric charge of $x$. Hence $m(x)$ is a twodimensional vector.

Let us define in this case:

$$F(a,b) = <m(a),m(b)>/log(k(a,b))$$ where we want to determine $k(a,b)$ in such a way, that (*) is fullfilled

$$F_N(a,b) = <m_N(a),m_N(b)>/log(k_N(a,b)), <m_N(a),m_N(b)>:= m_N(a)*m_N(b), k_N(a,b) := exp(-|x_a-x_b|^2/G)$$

, G = Gravity constant

$$F_C(a,b) = <m_C(a),m_C(b)>/log(k_C(a,b)), <m_C(a),m_C(b)>:= m_C(a)*m_C(b), k_C(a,b) := exp(-|x_a-x_b|^2/C)$$

,C = Coulombs constant

Using (*) and solving for $k(a,b)$ we find that for $a!=b$:

$$k(a,b) = exp( <m(a),m(b)>/( F_C(a,b)+F_N(a,b) ) )$$

and $k(a,a) := 1$

The mathematical question, which comes to my mind is this:

Why should the kernel defined by k(a,b) as above, be positive definite?

I have tested this idea with some Sagemath code empirically, but am not sure why it always should be positive definite, if it is at all:

def k1(a,b):
    G = 1
    return exp(-abs(a-b)**2/G)

def k2(a,b):
    C = 2
    return exp(-abs(a-b)**2/C)

def m1(a):
    return len(prime_divisors(a))+1

def m2(a):
    return sum(valuation(a,p) for p in prime_divisors(a))+1


def km(a,b):
    return m1(a)*m1(b)+m2(a)*m2(b)

def kk(a,b):
    if a==b:
        return 1
    return exp((km(a,b))/(m1(a)*m1(b)/log(k1(a,b))+m2(a)*m2(b)/log(k2(a,b))))

for n in range(1,100):
    M=(matrix([[kk(a,b) for a in range(1,n+1)] for b in range(1,n+1)],ring=RDF))
    if not M.is_positive_definite():
        print(n,M.eigenvalues())
    else:
        print(n)

    

Here is a Sagecell link for the code above, in case you do want to test it online: https://sagecell.sagemath.org/?z=eJx9kTFvgzAQhfdI_AePZ0JCYKzElKqZsnSNqshWDnQKHMg2UX5-DXFSV6j1YBlz3713zxesxbUAlWn5lqyEXwdRieJxNOhGwwLvA2yUtqA2WqZpmR9kskpWlwktY3Tv0fJ_dP9CO6_6BEN1iwyDoQ7PF7qR7Y3npFwXL6RcIHbs4KbaUTnq2XsZpKh7IwZBLP5qFax3sfXQbjaV-l3L9ayW-l3_jHuNGaqFqiodvqIuy_ggqMkcIom87RsI6f_Se_x4ZOvXJD9NxdNURnGDUGTFbvc0cqygU87QHU6nYHFOQcX1vC7k13ytF9eZIW6qz_ePSSuMxr0Txy3Z89BbcnTzSWJNTA5BRiP7kNkBZ8ctUoM8vQVaePbB1uKyWH4D1sio3w==&lang=sage&interacts=eJyLjgUAARUAuQ==